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Nonlinear Differential Equation in Anti-aging Test of Polymer Nanomaterials

Publicado en línea: 15 Jul 2022
Volumen & Edición: AHEAD OF PRINT
Páginas: -
Recibido: 14 Feb 2022
Aceptado: 26 Apr 2022
Detalles de la revista
License
Formato
Revista
eISSN
2444-8656
Primera edición
01 Jan 2016
Calendario de la edición
2 veces al año
Idiomas
Inglés
Introduction

Polymers, fiber composite materials, biomaterials, etc., are widely used in aerospace, vehicle engineering, civil engineering, bioengineering, and materials engineering. Composite materials often exhibit significant nonlinear viscoelastic properties under load. The nonlinear vibration of viscoelastic structures has attracted more and more attention from scholars at home and abroad. Some scholars have studied the periodic and chaotic motion of viscoelastic rods under the action of simple harmonic excitation [1]. Some scholars have studied viscoelastic Timoshenko beams. They derive the equations of motion of the system from assumed constitutive relations. At the same time, the system's nonlinear behavior is analyzed by numerical calculation. Some scholars have studied viscoelastic transmission belt lateral vibration's characteristics and chaotic dynamic behavior. Some scholars have studied the nonlinear vibration of carbon nano-reinforced composite sandwich cantilever beams. According to the Kelvin-Voigt model, they established the dynamic governing equations of the viscoelastic cantilever beam. At the same time, the stability of the system is studied by multiscale analysis and experiments. Some scholars have studied the lateral vibration of piles under nonlinear elastic and linear viscous conditions. Some scholars have studied the nonlinear vibration of simply supported viscoelastic beams with lumped masses at the ends [2]. Some scholars have studied the nonlinear dynamics, bifurcation, and chaotic dynamic responses of simply supported piezoelectric composite laminated beams under axial and lateral loads combined. Some scholars have analyzed the lateral motion of viscoelastic beams under the excitation of axial load [3]. They obtained the governing equations through viscoelastic constitutive relations. Some scholars have studied the nonlinear vibration of the isotropic viscoelastic layer and cantilever beam. Some scholars have studied the nonlinear vibration of variable-speed rotating viscoelastic beams. Some scholars have analyzed the bifurcation and chaotic characteristics of parametrically excited viscoelastic transmission belts.

Nonlinear constitutive relations often lead to complex dynamics governing equations. Our further analysis of the system's stability, bifurcation, and chaotic motion adds to the difficulty [4]. The experimental modeling method provides a way for the viscoelastic material to establish a mathematical model which is convenient for nonlinear dynamic analysis. We use ABS resin as the base material. At the same time, we made a series of composite samples with rutile nano-titanium dioxide filled with 1%-10%. This paper establishes an experimental system for parametric excitation of nonlinear vibrating beams [5]. The experimental setup is the lateral vibration of a viscoelastic simply supported beam subjected to the axial excitation force and applied with controllable dry friction damping. The nonlinear dynamics governing equations of viscoelastic parametric beams are obtained through experimental data and incremental harmonic balance nonlinear identification. x+ω02x+2μx+α1x2+α2x3+α3x3+α4x2x+α5xx2[ f0+fcos(Ωt) ]x=0 \matrix{ {x + \omega _0^2 + 2\mu x + {\alpha _1}{x^2} + {\alpha _2}{x^3} + {\alpha _3}{x^3} + } \hfill \cr {{\alpha _4}{x^2}x + {\alpha _5}x{x^2} - \left[ {{f_0} + f\cos \left( {\Omega t} \right)} \right]x = 0} \hfill \cr } Stability and bifurcation properties of our equation (1) solution at 1/2 subharmonic resonance [6]. We analyze the system's amplitude-frequency response, the solution's stability, and the bifurcation properties using a multiscale approach. At the same time, we use numerical simulation to verify the theoretical analysis results.

Experimental Modeling

We establish the dynamic governing equations of viscoelastic composite beams employing experimental modeling [7]. The article starts with the preparation of materials. At the same time, we prepared six different composite beam series samples according to the composition ratio of ABS and rutile nano-titanium dioxide (Table 1).

Composition ratio of ABS and rutile nano-TiO2

NO. 0 (high temperature) NO. 1 NO. 2 NO. 3 NO. 4 NO. 5
ABS/% 100 100 99 97 95 90
Nano TiO2/% 0 0 1 3 5 10

The experiment uses ABS material as the base material. Filled with other nanoscale components to modify ABS materials [8]. In this way, composite materials with corresponding functions can be obtained. Rutile nano titanium dioxide is widely used in plastic, rubber, and functional fiber products. It can improve the product's antiaging ability, anti-pulverization ability, weather resistance, and product strength. At the same time, it can maintain the product's color luster and prolong the product's service life.

In this paper, the experimental data of No. 3 material are used for parameter identification. Then use the experimental data of other materials for model verification. In this way, a dynamic governing equation suitable for nonlinear parametric vibration of a class of nanocomposite materials can be obtained. The article first establishes an experimental system corresponding to the model. The dynamic properties of viscoelastic composite beams under parametric excitation are studied [9]. Therefore, the experimental device is fixed at one end and slid at the other. In the experiment, we need to eliminate the influence of gravity. We will arrange the buckling beam longitudinally (Figure 1). The frequency modulation range of the system excitation frequency is between 0Hz-200Hz. The excitation amplitude f is controlled by adjusting the voltage. The nonlinear damping of the system is produced by the dry friction structure arranged at the sliding end. Force devices and force sensors control the nonlinear damping of the system. The lateral vibration response of the beam is measured by an accelerometer attached to the middle of the beam. Material No. 3 was chosen as the basic material for experimental modeling. Its size is 200mm×20mm×1.5mm. The mass is 7.3g. The first-order natural frequency is 6.5Hz.

Figure 1

Schematic diagram of the experimental setup

The dynamic model of the experimental system under the single-mode approximation is a nonlinear ordinary differential equation with parametric excitation terms. x¨+ω02x+cx˙+N(x,x˙)+x[ p0+p1cos(Ωt) ]=0 \ddot x + \omega _0^2x + c\dot x + N\left( {x,\,\dot x} \right) + x\left[ {{p_0} + {p_1}\cos \left( {\Omega t} \right)} \right] = 0 Therefore, when considering the dynamic model of the ABS-TiO2 nano-viscoelastic material beam, we need to assume that the nonlinear term of the system is a 3rd-degree polynomial. Then the dynamic model of the system is set to the following form: x¨+α0x+α1x˙+α2x2+α3xx˙+α4x˙2+α5x3+α6x2x˙+α7xx˙2+α8x˙3+α9cos(Ωt+φ0)x=0 \matrix{ {\ddot x + {\alpha _0}x + {\alpha _1}\dot x + {\alpha _2}{x^2} + {\alpha _3}x\dot x + {\alpha _4}{{\dot x}^2} + } \hfill \cr {{\alpha _5}{x^3} + {\alpha _6}{x^2}\dot x + {\alpha _7}x{{\dot x}^2} + {\alpha _8}{{\dot x}^3} + {\alpha _9}\,\cos \left( {\Omega t + {\varphi _0}} \right)x = 0} \hfill \cr } αi(i = 0, ⋯, 9) is a parameter that needs to be identified through experimental data and nonlinear parameter identification theory. We measure the vibration response data of the beam from the experimental system and apply the incremental harmonic balance nonlinear identification method to identify the parameter αi(i = 0, ⋯, 9) in equation (3). Substitute the above parameters into equation (3) to solve the response x(t). The results are then compared with the experimentally measured response results. Comparing the influence of various nonlinear terms in equation (3) on the identification results shows that the simulation results are in good agreement with the experimental results when xx˙ x\dot x is not included in equation (3) (Fig. 2). Otherwise, the solution of the equation will show decay to zero or infinity. However, whether the x2 term exists in the equation does not affect the recognition result. In any case, the equation can get good results [10]. But other models of material keeping item x2 will get better results. so the optimal dynamic control equation of viscoelastic beam under parametric excitation is: x¨+α0x+α1x˙+α2x2+α4x˙2+α5x3+α6x2x˙+α7xx˙2+α8x˙3+α9cos(Ωt+φ0)x=0 \matrix{ {\ddot x + {\alpha _0}x + {\alpha _1}\dot x + {\alpha _2}{x^2} + {\alpha _4}{{\dot x}^2} + {\alpha _5}{x^3} + {\alpha _6}{x^2}\dot x + {\alpha _7}x{{\dot x}^2} + } \hfill \cr {{\alpha _8}{{\dot x}^3} + {\alpha _9}\,\cos \left( {\Omega t + {\varphi _0}} \right)x = 0} \hfill \cr } Where α0 is related to the natural frequency and initial conditions. α1 is linear damping. α2 and α5 are nonlinear stiffnesses. α4, α6, α7 and α8 are related to nonlinear factors such as dry friction, internal damping, etc., α9 is the parameter excitation amplitude.

Figure 2

Comparison of the recognition results without xx˙ x\dot x in the model with the measured signal response and phase diagram

At this point, we verify the applicability of model (4). We use experimental data measured on beams of materials 1, 2, 4, and 5. The article compares the numerical simulation results with the experimental results (Figure 3). It can be seen from the figure that the simulation results of the model are in good qualitative and quantitative agreement with the measured results. Model (4) is well validated.

Figure 3

Comparison and verification of experimental results and simulation results of materials with different proportions

Solution stability

We take the No. 3 nanocomposite beam as an example to study the nonlinear dynamic characteristics of the beam under parametric excitation [11]. Therefore, we can write the governing equation (4) in the following form: x¨+ω02x+2μx˙+α1x˙2+α2x3+α3x˙3+α4x2x˙+α5xx˙2[ f0+fcos(Ωt) ]x=0 \matrix{ {\ddot x + \omega _0^2x + 2\mu \dot x + {\alpha _1}{{\dot x}^2} + {\alpha _2}{x^3} + {\alpha _3}{{\dot x}^3} + } \hfill \cr {{\alpha _4}{x^2}\dot x + {\alpha _5}x{{\dot x}^2} - \left[ {{f_0} + f\,\cos \left( {\Omega t} \right)} \right]x = 0} \hfill \cr } Where μ is the linear damping coefficient. ω0 is the natural frequency of the system. f0 and f are related to the parametric excitation, and αi(i = 0, ⋯, 5) is the nonlinear parameter of the system.

In the case of 1/2 subharmonic resonance, the system will have response and bifurcation characteristics. Suppose Ω2/4=ω02+εσ0 {\Omega ^2}/4 = \omega _0^2 + \varepsilon {\sigma _0} . Here σ0 0 is the tuning parameter. We substitute ω02=Ω2/4εσ0 \omega _0^2 = {\Omega ^2}/4 - \varepsilon {\sigma _0} into equation (5). At the same time, we use the multiscale method (5) to write: x¨+Ω24x+ε [ σx+2μ x˙+α1x˙2+α2x3+α3x˙3+α4x2x˙+α5xx˙2fcos(Ωt)x ]=0 \matrix{ {\ddot x + {{{\Omega ^2}} \over 4}x + \varepsilon \left[ { - \sigma x + 2\mu } \right.\dot x + {\alpha _1}{{\dot x}^2} + {\alpha _2}{x^3} + {\alpha _3}{{\dot x}^3} + } \hfill \cr {\left. {{\alpha _4}{x^2}\dot x + {\alpha _5}x{{\dot x}^2} - f\,\cos \left( {\Omega t} \right)x} \right] = 0} \hfill \cr } Where σ = σ0 + f0. Suppose the first-order asymptotic solution of equation (6) is of the form: x(t,ε)=v0(t1,t2)+εv1(t1,t2)+O(ε2) x\left( {t,\varepsilon } \right) = {v_0}\left( {{t_1},{t_2}} \right) + \varepsilon {v_1}\left( {{t_1},{t_2}} \right) + O\left( {{\varepsilon ^2}} \right) Where ti = ɛi−1t (i = 1, ⋯), ɛ 1, Di=ti {D_i} = {\partial \over {\partial {t_i}}} , DiDj = Di,j

We substitute equation (7) into equation (6). We equalize the coefficients of the same power on both sides of the equation to obtain the following system of differential equations: D1,1(v0)+14Ω2v0=0 {D_{1,1}}\left( {{v_0}} \right) + {1 \over 4}{\Omega ^2}{v_0} = 0 D12(v1)+14Ω2v1=σv02D1,2(v0)2μD1(v0)α4v02D1(v0)α1D1(v0)2+v0fcos(Ωt1)α5v0D1(v0)2α2v03α3D1(v0)3 \matrix{ {D_1^2\left( {{v_1}} \right) + {1 \over 4}{\Omega ^2}{v_1} = \sigma {v_0} - 2{D_{1,2}}\left( {{v_0}} \right) - } \hfill \cr {2\mu {D_1}\left( {{v_0}} \right) - {\alpha _4}v_0^2{D_1}\left( {{v_0}} \right) - {\alpha _1}{D_1}{{\left( {{v_0}} \right)}^2} + } \hfill \cr {{v_0}f\,\cos \left( {\Omega {t_1}} \right) - {\alpha _5}{v_0}{D_1}{{\left( {{v_0}} \right)}^2} - {\alpha _2}v_0^3 - {\alpha _3}{D_1}{{\left( {{v_0}} \right)}^3}} \hfill \cr } The solution to equation (8) is: v0=A(t2)e12iΩt1+A(t2)e¯12iΩt1 {v_0} = A\left( {{t_2}} \right){e^{{1 \over 2}i\Omega {t_1}}} + {\overline {A\left( {{t_2}} \right)e} ^{ - {1 \over 2}i\Omega {t_1}}} Here A(t2) is the complex conjugate of A(t2)¯ \overline {A\left( {{t_2}} \right)} . For long term terms to not appear in the solution, then there must be: dA(t2)dt2=iσA(t2)Ω38Ω2α3A(t2)2A(t2)¯+14iΩα5A(t2)2A(t2)¯μA(t2)12α4A(t2)2A(t2)¯ifA(t2)¯2Ω+3iα2A(t2)2A(t2)¯Ω \matrix{ {{{dA\left( {{t_2}} \right)} \over {d{t_2}}} = - {{i\sigma A\left( {{t_2}} \right)} \over \Omega } - {3 \over 8}{\Omega ^2}{\alpha _3}A{{\left( {{t_2}} \right)}^2}\,\overline {A\left( {{t_2}} \right)} + {1 \over 4}i\Omega {\alpha _5}A{{\left( {{t_2}} \right)}^2}\overline {A\left( {{t_2}} \right)} - \mu A\left( {{t_2}} \right) - } \hfill \cr {{1 \over 2}{\alpha _4}A{{\left( {{t_2}} \right)}^2}\overline {A\left( {{t_2}} \right)} - {{if\overline {A\left( {{t_2}} \right)} } \over {2\Omega }} + {{3i{\alpha _2}A{{\left( {{t_2}} \right)}^2}\overline {A\left( {{t_2}} \right)} } \over \Omega }} \hfill \cr } Suppose: A(t2)=12a(t2)eiθ(t2) A\left( {{t_2}} \right) = {1 \over 2}a\left( {{t_2}} \right){e^{i\theta \left( {{t_2}} \right)}} We substitute equation (12) into equation (11) and separate the real and imaginary parts. Simplify to get the average equation in the polar coordinate form under the first approximation as { Ωda(t2)dt2=18α4a(t2)3Ω332α3a(t2)3Ω3a(t2)μΩ12fa(t2)sin[ 2θ(t2) ]Ωa(t2)dθ(t2)dt2=116α5a(t2)3Ω2+34α3a(t2)3σa(t2)12fa(t2)cos[ 2θ(t2) ] \left\{ {\matrix{ {\Omega {{da\left( {{t_2}} \right)} \over {d{t_2}}} = - {1 \over 8}{\alpha _4}a{{\left( {{t_2}} \right)}^3}\Omega - {3 \over {32}}{\alpha _3}a{{\left( {{t_2}} \right)}^3}{\Omega ^3} - a\left( {{t_2}} \right)\mu \Omega - {1 \over 2}fa\left( {{t_2}} \right)\sin \left[ {2\theta \left( {{t_2}} \right)} \right]} \hfill \cr {\Omega a\left( {{t_2}} \right){{d\theta \left( {{t_2}} \right)} \over {d{t_2}}} = {1 \over {16}}{\alpha _5}a{{\left( {{t_2}} \right)}^3}{\Omega ^2} + {3 \over 4}{\alpha _3}a{{\left( {{t_2}} \right)}^3} - \sigma a\left( {{t_2}} \right) - {1 \over 2}fa\left( {{t_2}} \right)\cos \left[ {2\theta \left( {{t_2}} \right)} \right]} \hfill \cr } } \right. At this point, we discuss the steady-state solution of equation (13) and the variation of the system response with the parameter σ0, μ. Let dadt2=0 {{da} \over {d{t_2}}} = 0 , dθdt2=0 {{d\theta } \over {d{t_2}}} = 0 and eliminate θ to get the bifurcation response equation as: { [ 116(α4Ω+34α3Ω3)2+164(α5Ω2+12α2)2 ]α4+(34α3Ω4μ12α5Ω2σ+α4Ω2μ6α2σ)α2+4μ2Ω2+4σ2f2=0a=0 \left\{ {\matrix{ {\left[ {{1 \over {16}}{{\left( {{\alpha _4}\Omega + {3 \over 4}{\alpha _3}{\Omega ^3}} \right)}^2} + {1 \over {64}}{{\left( {{\alpha _5}{\Omega ^2} + 12{\alpha _2}} \right)}^2}} \right]{\alpha ^4} + } \hfill \cr {\left( {{3 \over 4}{\alpha _3}{\Omega ^4}\mu - {1 \over 2}{\alpha _5}{\Omega ^2}\sigma + {\alpha _4}{\Omega ^2}\mu - 6{\alpha _2}\sigma } \right){\alpha ^2} + } \hfill \cr {4{\mu ^2}{\Omega ^2} + 4{\sigma ^2} - {f^2} = 0} \hfill \cr {a = 0} \hfill \cr } } \right. Suppose: A1=116(α4Ω+34α3Ω3)2+164(α5Ω2+12α2)2;B=34α3Ω4μ12α5Ω2σ+α4Ω2μ6α2σ;C=4μ2Ω2+4σ2f2;Δ=B24A1C \matrix{ {{A_1} = {1 \over {16}}{{\left( {{\alpha _4}\Omega + {3 \over 4}{\alpha _3}{\Omega ^3}} \right)}^2} + {1 \over {64}}{{\left( {{\alpha _5}{\Omega ^2} + 12{\alpha _2}} \right)}^2};} \hfill \cr {B = {3 \over 4}{\alpha _3}{\Omega ^4}\mu - {1 \over 2}{\alpha _5}{\Omega ^2}\sigma + {\alpha _4}{\Omega ^2}\mu - 6{\alpha _2}\sigma ;} \hfill \cr {C = 4{\mu ^2}{\Omega ^2} + 4{\sigma ^2} - {f^2};\Delta = {B^2} - 4{A_1}C} \hfill \cr } Equation (14) can be abbreviated as: A1a4+Ba2+C=0;a=0 {A_1}{a^4} + B{a^2} + C = 0;\,a = 0 From this, the following different solutions can be obtained. There is A1 ≥ 0, here is assumed: A1>0,4α4+3α3Ω2>0,α5Ω2+12α2>0 {A_1} > 0,\,4{\alpha _4} + 3{\alpha _3}{\Omega ^2} > 0,\,{\alpha _5}{\Omega ^2} + 12{\alpha _2} > 0

When C < 0, there are: a1,2=0 {a_{1,2}} = 0 a3,4=B+B24A1C2A1 {a_{3,4}} = \sqrt {{{ - B + \sqrt {{B^2} - 4{A_1}C} } \over {2{A_1}}}}

When B < 0, C ≥ 0, Δ = B2 −4 A1C ≥ 0, there are: a1,2=0 {a_{1,2}} = 0 a3,4=B+B24A1C2A1 {a_{3,4}} = \sqrt {{{ - B + \sqrt {{B^2} - 4{A_1}C} } \over {2{A_1}}}} a5,6=BB24A1C2A1 {a_{5,6}} = \sqrt {{{ - B - \sqrt {{B^2} - 4{A_1}C} } \over {2{A_1}}}}

Other circumstances include: a1,2=0 {a_{1,2}} = 0

We judge the stability of the solution at 1/2 subharmonic resonance. At the same time, we transform the average equation (13) from polar form to Cartesian form: A(t2)=x+yi A\left( {{t_2}} \right) = x + yi Here x and y are real functions of t2. We substitute equation (21) into equation (11) and separate the real and imaginary parts from obtaining the average equation in Cartesian coordinates: dxdt2Ω=12α4Ωx3+σy12α4Ωxy238α3Ω3x33α2y312fy14α5Ω2yx23α2yx214α5Ω2y3μΩx38α3Ω3xy2dydt2Ω=14α5Ω2xy212fxσx38α3Ω3yx212α4Ωyx2+3α2xy2+14α5Ω2x338α3Ω3y312α4Ωy3μΩy+3α2x3 } \left. {\matrix{ {{{dx} \over {d{t_2}}}\Omega = - {1 \over 2}{\alpha _4}\Omega {x^3} + \sigma y - {1 \over 2}{\alpha _4}\Omega x{y^2} - } \hfill \cr {{3 \over 8}{\alpha _3}{\Omega ^3}{x^3} - 3{\alpha _2}{y^3} - {1 \over 2}fy - } \hfill \cr {{1 \over 4}{\alpha _5}{\Omega ^2}y{x^2} - 3{\alpha _2}y{x^2} - {1 \over 4}{\alpha _5}{\Omega ^2}{y^3} - \mu \Omega x - {3 \over 8}{\alpha _3}{\Omega ^3}x{y^2}} \hfill \cr {{{dy} \over {d{t_2}}}\Omega = {1 \over 4}{\alpha _5}{\Omega ^2}x{y^2} - {1 \over 2}fx - \sigma x - } \hfill \cr {{3 \over 8}{\alpha _3}{\Omega ^3}y{x^2} - {1 \over 2}{\alpha _4}\Omega y{x^2} + 3{\alpha _2}x{y^2} + } \hfill \cr {{1 \over 4}{\alpha _5}{\Omega ^2}{x^3} - {3 \over 8}{\alpha _3}{\Omega ^3}{y^3} - {1 \over 2}{\alpha _4}\Omega {y^3} - \mu \Omega y + 3{\alpha _2}{x^3}} \hfill \cr } } \right\} From the Jacobi matrix of equation (22), the Eigen equation corresponding to the zero solution can be obtained: λ2+2λμΩ+μ2Ω2+σ214f2=0 {\lambda ^2} + 2\lambda \mu \Omega + {\mu ^2}{\Omega ^2} + {\sigma ^2} - {1 \over 4}{f^2} = 0 The Eigen equation corresponding to the non-zero solution is: λ2+[ (12α4Ω+38α3Ω3)a2+2μΩ ]λ+(3α2σ38μα3Ω4+14α5σΩ212μα4Ω2)a24μ2Ω24σ2+f2=0 \matrix{ {{\lambda ^2} + \left[ {\left( {{1 \over 2}{\alpha _4}\Omega + {3 \over 8}{\alpha _3}{\Omega ^3}} \right){a^2} + 2\mu \Omega } \right]\lambda + } \hfill \cr {\left( {\matrix{ {3{\alpha _2}\sigma - {3 \over 8}\mu {\alpha _3}{\Omega ^4} + } \hfill \cr {{1 \over 4}{\alpha _5}\sigma {\Omega ^2} - {1 \over 2}\mu {\alpha _4}{\Omega ^2}} \hfill \cr } } \right){a^2} - 4{\mu ^2}{\Omega ^2} - 4{\sigma ^2} + {f^2} = 0} \hfill \cr } According to the different solutions of the bifurcation response equation (14) and the Eigen equation (23), the stable region where the solution of (24) can be obtained is shown in Fig. 4. The singularity of the average equation (13) and the periodic solution of the governing equation (6) have the same stable region on the parametric plane (σ0, μ).

Figure 4

Steady-state solution stability region

Zero solution stability analysis

We analyze the zero-solution stability in different regions according to the zero-solution Eigen equation (23). At this point, we get the following result:

When μ > 0 is shown in Figure 4. The zero solution is stable in the region I. In the region II is unstable.

When μ < 0, the eigenvalues of the zero solution have at least one eigenvalue whose real part is greater than zero. so the zero solution is always unstable.

Non-zero solution stability analysis

According to the non-zero solution characteristic root structure of Eq. (24), it can be known that the stability condition of the non-zero solution is: (12α4Ω+38α3Ω3)a2+2μΩ>0 \left( {{1 \over 2}{\alpha _4}\Omega + {3 \over 8}{\alpha _3}{\Omega ^3}} \right){a^2} + 2\mu \Omega > 0 (3α2σ38μα3Ω4+14α5σΩ212μα4Ω2)a24μ2Ω24σ2+f2>0 \left( {3{\alpha _2}\sigma - {3 \over 8}\mu {\alpha _3}{\Omega ^4} + {1 \over 4}{\alpha _5}\sigma {\Omega ^2} - {1 \over 2}\mu {\alpha _4}{\Omega ^2}} \right){a^2} - 4{\mu ^2}{\Omega ^2} - 4{\sigma ^2} + {f^2} > 0 The stable and unstable regions of non-zero solutions can be obtained by stability analysis as follows:

When μ > 0 is shown in Figure 4. The non-zero solution (16) is stable in region II, and the non-zero solution (18) is stable in region VIII. The non-zero solution of equation (19) is not stable in region VIII.

When μ < 0, the non-zero solution of equation (16) is unstable in region IV and stable in region V. The non-zero solution (18) is stable in region VI but unstable in region VII. The non-zero solution of equation (19) is not stable at VI and VII.

Bifurcation Analysis

System stability varies with system parameters. At this time, the system exhibits different motion forms. Next, numerical simulation is used to analyze the influence of the parameter excitation amplitude f on the bifurcation behavior of the system [12]. The main parameter of the system (5) is ω0 = 1, Ω = 2, f0 = 0.1, μ = 0.05, αi = 0.1(i = 1, 2, 3, 4, 5). Fig. 5 is a graph of the maximum Lyapunov exponent concerning the parameter excitation amplitude f, and Fig. 6 is a bifurcation graph obtained by the Runge-Kutta numerical integration method concerning the parameter excitation amplitude f. The results show that the system exhibits rich, dynamic phenomena with parameter variation f.

1) When the excitation amplitude f ∈ (0,0.286), the system begins to move in a decaying motion due to the damping action. The system is finally at rest. 2) When the incentive effect is further strengthened, f ∈ (0.286,12), the system appears in periodic motion [13]. The system goes through period 1→period 2→period 3. 3) From Figure 5 and Figure 6, it can be seen that the system will change from period 3 to period 4 with the increase of excitation amplitude f. 4) Then the system undergoes period-doubling bifurcation and enters chaos state. When f ∈ (13.0445, 15.3655), the maximum Lyapunov exponent in Figure 5 is positive. This indicates that the system is in a chaotic state at this time.

Figure 5

Maximum Lyapunov exponent plot of the system

Figure 6

Bifurcation diagram of the system

Conclusion

We analyze the dynamics of the system. The influence of the linear damping coefficient on the system's stability under 1/2 subharmonic resonance is discussed. At this time, the amplitude-frequency response of the system and the stable region in the (μ, σ0) plane are obtained. Numerical simulation is used to analyze the influence of parameter excitation on bifurcation behavior. At the same time, we discussed the path of the system to chaos. It is found that chaos is entered through period-doubling bifurcation. At the same time, we found that the system has paroxysmal chaos. This provides a theoretical basis for better utilization of nanocomposites.

Figure 1

Schematic diagram of the experimental setup
Schematic diagram of the experimental setup

Figure 2

Comparison of the recognition results without 



xx˙
x\dot x


 in the model with the measured signal response and phase diagram
Comparison of the recognition results without xx˙ x\dot x in the model with the measured signal response and phase diagram

Figure 3

Comparison and verification of experimental results and simulation results of materials with different proportions
Comparison and verification of experimental results and simulation results of materials with different proportions

Figure 4

Steady-state solution stability region
Steady-state solution stability region

Figure 5

Maximum Lyapunov exponent plot of the system
Maximum Lyapunov exponent plot of the system

Figure 6

Bifurcation diagram of the system
Bifurcation diagram of the system

Composition ratio of ABS and rutile nano-TiO2

NO. 0 (high temperature) NO. 1 NO. 2 NO. 3 NO. 4 NO. 5
ABS/% 100 100 99 97 95 90
Nano TiO2/% 0 0 1 3 5 10

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